From: asimov@nas.nasa.gov (Daniel A. Asimov)
Newsgroups: sci.math.research
Subject: Re: Realizability in R^3 of Simplicial Orientable Surfaces
Date: Tue, 20 Jun 1995 23:28:43 GMT
In article asimov@nas.nasa.gov (Daniel A. Asimov) writes:
>In the book "Convex Polytopes" by Branko Grunbaum (1967), the following
>conjecture is described as neither proved nor disproved:
>
> Given any finite simplicial complex K whose underlying topological space is a
> compact orientable surface without boundary, then K can be realized as an
> embedded piecewise linear subset of R^3.
>
>(An easy argument shows that any such K, even if non-orientable, can be so
>realized in R^5.)
>
>
>QUESTIONS:
>
>1. Is this conjecture still unresolved?
>
>2. What about the non-orientable case? Is it possible that all such
> non-orientable K can be so realized in R^4 ?
>
>Daniel Asimov
>
>Mail Stop T27A-1
>NASA Ames Research Center
>Moffett Field, CA 94035-1000
>
>asimov@nas.nasa.gov
>(415) 604-4799 w
>(415) 604-3957 fax
>
>[I assume that Grunbaum means an embedding which is linear on each
>simplex; if the embedding is merely PL on each simplex then the
>question is trivial. -Greg]
P.S. Yes, indeed, I mean that each simplex of K is embedded affinely in R^3.
Grunbaum is not to blame; the above is my own paraphrase of his statement.
--Dan